Instability of rarefaction shocks in systems of conservation laws

It is well-known that rarefaction shocks are unstable solutions of nonlinear hyperbolic conservation laws. Indeed, for scalar equations rarefaction shocks are unstable in the class of smooth solutions, but for systems one can only say in general that rarefaction shocks are unstable in the larger class of weak solutions. (Here unstable refers to a lack of continuous dependence upon perturbations of the initial data.) Since stability in the class of weak solutions is not well understood, ([T, TE]), entropy considerations have played a leading role in ruling out shocks that violate the laws of physics. However, for non-strictly hyperbolic systems the analogy with the equations of gas dynamics breaks down, and general entropy or admissibility criteria for the variety of shocks which appear, (see, e.g., [IMPT]), are not known. In this paper we address the question of when the instability of a shock can be demonstrated within the class of smooth solutions alone. We show by elementary constructions that this occurs whenever there exists an alternative solution to the Riemann problem with the same shock data which consists entirely of rarefaction waves and contact discontinuities with at least one non-zero rarefaction wave. We show that for 2×2 strictly hyperbolic, genuinely nonlinear systems the condition is both necessary and sufficient. We show too that for the full 3×3 (Euler) equations of gas dynamics with polytropic equations of state, rarefaction shocks of moderate strength are unstable in the class of smooth solutions if and only if the adiabatic gas constant satisfies 1 < < 5/3 (see Theorem 8). More precisely, there is a constant y _*, 0 < y _* < 1, depending only on , such that if y _* p _lp _rp _l for 1-shocks, and if y _* p _rP _lp _r for 3-shocks (where p _r and p _l denote the pressures on both sides of the rarefaction shock), then the shock is unstable if and only if 1 < < 5/3. Thus for such shocks, the theory of the Riemann problem for polytropic gases in the range 1 < < 5/3 can be rigorously developed with a knowledge of the smooth solutions alone by using stability under smoothing as an admissibility criterion, rather than by using the classical entropy inequalities.     

Constant rate production of geothermal fluid from a two-phase vertical column I: Theory

Previous theoretical results on geothermal two-phase flows in porous media are extended and applied to the case of withdrawal of fluid at a constant rate from a vertical column. Dimensional considerations show that pressure and saturation behaviour is controlled by a single parameter which is the ratio of the withdrawal speed to buoyancy speed. For large flows (large) fluid withdrawal is a mining process, and a vapour dominated zone spreads out from the production level. Production enthalpies tend towards steam values. However if is small then gravity dominates, and buoyancy forces can lead to the formation of a steam bubble which escapes from the production boundary and rises towards the surface. Production enthalpy may then remain at the liquid value over long periods. In addition certain saturation ranges at the sink may be forbidden as a consequence of the constant rate boundary condition. Then saturation shocks will form at the production boundary and travel out from the sink. Internally generated shocks may also occur. Pressure and saturation response to a steady withdrawal of fluid is more complicated than in a two-phase gravity-free situation. Since gravity is an essential component of even horizontal two-phase flow this suggests that two-phase studies which ignore the role of gravity may be too simplistic.     

Shocks in local supersonic zones

The results of numerical integration of the Euler equations governing two-dimensional and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with local supersonic zones are presented. The subject of the study is the formation of shocks closing local supersonic zones. The flow in the vicinity of the initial point of the closing shock is calculated on embedded, successively refined grids with an accuracy much greater than that previously achieved. The calculations performed, together with the analysis of certain controversial issues, leave no doubt that it is the intersection of C ^?-characteristics proceeding from the sonic line inside the supersonic zone that is responsible for the closing shock formation.     

Jordan–Cattaneo waves: Analogues of compressible flow

We review work of Jordan on a hyperbolic variant of the Fisher–KPP equation, where a shock solution is found and the amplitude is calculated exactly. The Jordan procedure is extended to a hyperbolic variant of the Chafee–Infante equation. Extension of Jordan’s ideas to a model for traffic flow are also mentioned. We also examine a diffusive susceptible–infected (SI) model, and generalizations of diffusive Lotka–Volterra equations, including a Lotka–Volterra–Bass competition model with diffusion. For all cases we show how a Jordan–Cattaneo wave may be analysed and we indicate how to find the wavespeeds and the amplitudes. Finally we present details of a fully nonlinear analysis of acceleration waves in a Cattaneo–Christov poroacoustic model.     

Experimental investigation on tunnel sonic boom

Upon the entrance of a high-speed train into a relatively long train tunnel, compression waves are generated in front of the train. These compression waves subsequently coalesce into a weak shock wave so that a unpleasant sonic boom is emitted from the tunnel exit. In order to investigate the generation of the weak shock wave in train tunnels and the emission of the resulting sonic boom from the train tunnel exit and to search for methods for the reduction of these sonic booms, a 1300 scaled train tunnel simulator was constructed and simulation experiments were carried out using this facility.In the train tunnel simulator, an 18 mm dia. and 200 mm long plastic piston moves along a 40 mm dia. and 25 m long test section with speed ranging from 60 to 100 m/s. The tunnel simulator was tilted 8° to the floor so that the attenuation of the piston speed was not more than 10 % of its entrance speed. Pressure measurements along the tunnel simulator and holographic interferometric optical flow visualization of weak shock waves in the tunnel simulator clearly showed that compression waves, with propagation, coalesced into a weak shock wave. Although, for reduction of the sonic boom in prototype train tunnels, the installation of a hood at the entrance of the tunnels was known to be useful for their suppression, this effect was confirmed in the present experiment and found to be effective particularly for low piston speeds. The installation of a partially perforated wall at the exit of the tunnel simulator was found to smear pressure gradients at the shock. This effect is significant for higher piston speeds. Throughout the series of train tunnel simulator experiments, the combination of both the entrance hood and the perforated wall significantly reduces shock overpressures for piston speeds ofu _p ranging from 60 to 100 m/s. These experimental findings were then applied to a real train tunnel and good agreement was obtained between the tunnel simulator result and the real tunnel measurements.     

Modeling plastic shocks in periodic laminates with gradient plasticity theories

Steady plastic shocks generated by planar impact on metal-polymer laminate composites are analyzed in the framework of gradient plasticity theories. The laminate material has a periodic structure with a unit cell composed of two layers of different materials. First- and second-order gradient plasticity theories are used to model the structure of steady plastic shocks. In both theories, the effect of the internal structure is accounted for at the macroscopic level by two material parameters that are dependent upon the layer's thickness and the properties of constituents. These two structure parameters are shown to be uniquely determined from experimental data. Theoretical predictions are compared with experiments for different cell sizes and for various shock intensities. In particular, the following experimental features are well-reproduced by the modeling:

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the shock width is proportional to the cell size;

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the magnitude of strain rate is inversely proportional to cell size and increases with the amplitude of applied stress following a power law.

While these results are equally described by both the plasticity theories, the first gradient plasticity approach seems to be favored when comparing the structure of the shock front to the experimental data.     

Computational analysis of dense gas shock tube flow

Nonclassical phenomena associated with the classical dynamics of real gases in a conventional shock tube are studied. A TVD predictor-corrector (TVD-MacCormack) scheme with reflective endwall boundary conditions is used for the one-dimensional Euler equations to simulate the evolution of the wave field of a van der Waals gas. Depending upon the initial conditions of the gas, wave fields are produced that contain nonclassical phenomena such as expansion shocks, composite waves, splitting shocks, etc. In addition, the interactions of waves reflected from the endwalls produce both classical and nonclassical phenomena. Wave field evolution is depicted using plots of the flow variables at specific times and withx-t diagrams.     

Criteria on Contractions for Entropic Discontinuities of Systems of Conservation Laws

We study the contraction properties (up to shift) for admissible Rankine–Hugoniot discontinuities of \({n\times n}\) systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in (Serre and Vasseur, J l’Ecole Polytech 1, 2014), using the spatially inhomogeneous pseudo-distance introduced in (Vasseur, Contemp Math AMS, 2013). Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. None of the results involve any smallness condition on the initial perturbation or on the size of the shock.     

Use of flows with straight sonic line in nozzles with corners

The optimal scheme of a Laval nozzle is discussed. In the case of a profiled nozzle with a corner it is possible to use in the region of mixed flow both flows of general form with curvilinear sonic line as well as the special case when the sonic line is straight. It is shown that the latter alternative is preferable: when the supersonic part of the profile is determined by the simple wave method, the velocity at the wall increases monotonically and the flow does not contain shock waves. In contrast, in nozzles with curvilinear sonic line, a section in which the velocity decreases is formed immediately behind the corner, which can lead to boundary layer separation. In addition, for values of the supersonic velocity at the nozzle exit near the velocity of sound it is proved that the characteristics of the simple wave intersect in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–170, January–February, 1981.     

Sonic discontinuities in a relaxing gas

Following Elcrat [1] the phenomena associated with the sonic discontinuities in non-equilibrium gasdynamics have been studied here. The sonic wave in non-equilibrium gaseous medium propagates with the frozen speed of sound. The magnitude of discontinuities of the first derivatives of flow quantities in the unsteady flow of relaxing gas are shown to satisfy Riccatti equations along the orthogonal trajectories of surface S(t). In order to integrate them in full generality, they are transformed to an equation along the bicharacteristic curve in the characteristic manifold S(t). These equations have been solved completely. The criteria for decay or blow up of sonic discontinuities are given and the particular cases of plane and spherical waves existing in an ideal dissociative gasdynamics have been discussed. In the case of planewave for uniform propagation, it is shown that the dissociating character of the gas is to decrease the critical time. Other cases of shock formation have been studied in detail.     

On the distance to blow-up of acceleration waves in random media

We study the effects of material spatial randomness on the distance to form shocks from acceleration waves, , in random media. We introduce this randomness by taking the material coefficients and – that represent the dissipation and elastic nonlinearity, respectively, in the governing Bernoulli equation – as a stochastic vector process. The focus of our investigation is the resulting stochastic, rather than deterministic as in classical continuum mechanics studies, competition of dissipation and elastic nonlinearity. Quantitative results for are obtained by the method of moments in special simple cases, and otherwise by the method of maximum entropy. We find that the effect of even very weak random perturbation in and may be very significant on . In particular, the full negative cross-correlation between and $ results in the strongest scatter of , and hence, in the largest probability of shock formation in a given distance x. Received November 6, 2001 / Published online September 4, 2002 Dedicated to Professor Ingo Müller on the occasion of his 65th birthday Communicated by Kolumban Hutter, Darmstadt     

Multidimensional Shock Interaction for a Chaplygin Gas

A Chaplygin gas is an inviscid, compressible fluid in which the acoustic fields are linearly degenerate. We analyze the multidimensional shocks in such a fluid, which turn out to be sonic. Two shocks in general position interact rather simply. We investigate several two-dimensional Riemann problems and prove the existence of a unique solution. Among them is the supersonic reflection of a planar shock against a wedge; we remark that the solution cannot be a Mach reflection, contrary to what happens for other gases, and that there always exists a solution in the form of a regular reflection.     

Nonstationary motion of a shell on the surface of a heavy fluid

A three-dimensional nonstationary problem of vibrations of a flexible shell moving on the surface of an ideal heavy fluid. The forces due to surface tension are ignored. The problem is formulated in the space of the acceleration potential. The potential of the pulsating source is found by solving the Euler equation and the continuity equation taking into account the free-surface conditions (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions on the surface of the shell. Formulas for determining the shape of gravity waves on the fluid surface and the natural frequencies of vibrations of the shell are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 66–75, July–August, 2009.     

Semi-hyperbolic Patches of Solutions to the Two-dimensional Euler Equations

We construct semi-hyperbolic patches of solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional Euler equations. This type of solution appears in the transonic flow over an airfoil and Guderley reflection, and is common in the numerical solutions of Riemann problems.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

Properties of Elastic Waves in a Non-Newtonian (Maxwell) Fluid-Saturated Porous Medium

The present study investigates novelties brought into the classic Biot's theory of propagation of elastic waves in a fluid-saturated porous solid by inclusion of non-Newtonian effects that are important, for example, for hydrocarbons. Based on our previous results (Tsiklauri and Beresnev, 2001), we investigated the propagation of rotational and dilatational elastic waves by calculating their phase velocities and attenuation coefficients as a function of frequency. We found that the replacement of an ordinary Newtonian fluid by a Maxwell fluid in the fluid-saturated porous solid results in: (a) an overall increase of the phase velocities of both the rotational and dilatational waves. With the increase of frequency these quantities tend to a fixed, higher level, as compared to the Newtonian limiting case, which does not change with the decrease of the Deborah number . (b) The overall decrease of the attenuation coefficients of both the rotational and dilatational waves. With the increase of frequency these quantities tend to a progressively lower level, as compared to the Newtonian limiting case, as decreases. (c) Appearance of oscillations in all physical quantities in the deeply non-Newtonian regime.     

Kinetic Stabilization¶of a Nonlinear Sonic Phase Boundary

We consider a system arising in the study of phase transitions in elastodynamics – a system of two conservation laws, in a single space dimension. The system has two hyperbolic regions with an elliptic zone in between. A phase boundary is a strong discontinuity in a solution, with left and right states belonging to different hyperbolic regions. We call such a solution a phase wave. We first address the Riemann problem for initial states close to a fixed sonic phase wave, in the genuinely nonlinear case. This problem is naturally underdetermined. We propose two essentially different types of Reimann problems: a sonic one, which is smooth, and a kinetic one, which is only Lipschitz-continuous. Both problems are well posed owing to a shared stability condition that is of a purely sonic nature. In the kinetic case we prove the global existence of solutions to the Cauchy problem for initial data having small variation and close to a sonic kinetic wave. The crucial issue is the interaction of the phase boundary with a small wave of the same mode. The introduction of a pertinent quantity, called here detonation potential, ensures a balance between ingoing and outgoing waves. The proof is based on a Glimm-type scheme; we define a potential, which includes the detonation potential, along the strong discontinuity, and this potential controls the outbreak of unusual shocks. Accepted: June 9, 1999     

Radiative shock solutions with grey nonequilibrium diffusion

This study describes a semi-analytic solution of planar radiative shock waves with a grey nonequilibrium diffusion radiation model. The solution may be used to verify radiation-hydrodynamics codes. Comparisons are made with the equilibrium diffusion solutions of Lowrie and Rauenzahn (Shock Waves 16(6):445–453, 2007). The solution also gives additional insight into the structure of radiative shocks. Previous work has assumed that the material temperature reaches its maximum at the post-shock state of the embedded hydrodynamic shock (Zel’dovich spike). We show that in many cases, the temperature may continue to increase after the hydrodynamic shock and reaches its maximum at the isothermal sonic point. Also, a temperature spike may exist even in the absence of an embedded hydrodynamic shock. We also derive an improved estimate for the maximum temperature.